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InverseFourierCosTransform
✖
InverseFourierCosTransform

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✖

gives the symbolic inverse Fourier cosine transform of expr.

✖
InverseFourierCosTransform[expr,{ω1,ω2,…},{t1,t2,…}]

gives the multidimensional inverse Fourier cosine transform of expr.

Details and Options

  • The Fourier cosine transform is a particular way of viewing the Fourier transform without the need for complex numbers or negative frequencies.
  • Joseph Fourier designed his famous transform using this and the Fourier sine transform, and they are still used in applications like signal processing, statistics and image and video compression.
  • The inverse Fourier cosine transform of the frequency domain function is the time domain function for :
  • The inverse Fourier cosine transform of a function is by default defined as .
  • The multidimensional inverse Fourier cosine transform of a function is by default defined as or when using vector notation, (2/pi)^(n/2)int_(omega in TemplateBox[{}, PositiveReals]^n)F(omega) cos(omega t)domega.
  • Different choices of definitions can be specified using the option FourierParameters.
  • The integral is computed using numerical methods if the third argument, , is given a numerical value.
  • The asymptotic inverse Fourier cosine transform can be computed using Asymptotic.
  • There are several related Fourier transformations:
  • FourierTransforminfinite continuous-time functions (FT)
    FourierSequenceTransforminfinite discrete-time functions (DTFT)
    FourierCoefficientfinite continuous-time functions (FS)
    Fourierfinite discrete-time functions (DFT)
  • The inverse Fourier cosine transform is an automorphism in the Schwartz vector space of functions whose derivatives are rapidly decreasing and thus induces an automorphism in its dual: the space of tempered distributions. These include absolutely integrable functions, well-behaved functions of polynomial growth and compactly supported distributions.
  • Hence, InverseFourierCosTransform not only works with absolutely integrable functions on , but it can also handle a variety of tempered distributions such as DiracDelta to enlarge the pool of functions or generalized functions it can effectively transform.
  • The lower limit of the integral is effectively taken to be TemplateBox[{0, -}, Superscript], so that the inverse Fourier cosine transform of the Dirac delta function is equal to . »
  • The following options can be given:
  • AccuracyGoal Automaticdigits of absolute accuracy sought
    Assumptions $Assumptionsassumptions to make about parameters
    FourierParameters {0,1}parameters to define the inverse Fourier cosine transform
    GenerateConditions Falsewhether to generate answers that involve conditions on parameters
    PerformanceGoal$PerformanceGoalaspects of performance to optimize
    PrecisionGoal Automaticdigits of precision sought
    WorkingPrecision Automaticthe precision used in internal computations
  • Common settings for FourierParameters include:
  • {0,1}
    {1,1}
    {-1,1}
    {0,2Pi}
    {a,b}

Examples

open allclose all

Basic Examples  (6)Summary of the most common use cases

Compute the inverse Fourier cosine transform of a function:

In[1]:=1
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-gm7wa2
Out[1]=1

Plot the function and its inverse cosine transform:

In[2]:=2
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-5fci89
Out[2]=2

Inverse Fourier cosine transform of reciprocal square root:

In[1]:=1
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-28dp0t
Out[1]=1

For a different convention, change the parameters:

In[1]:=1
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-py5hxq
Out[1]=1

Inverse Fourier cosine transform of a Gaussian is another Gaussian:

In[1]:=1
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-fwkh1w
Out[1]=1

Plot both Gaussians:

In[2]:=2
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-treoee
Out[2]=2

Compute the inverse Fourier cosine transform of a multivariate function:

In[1]:=1
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-7gezt1
Out[1]=1

Plot the result:

In[2]:=2
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-5vrckt
Out[2]=2

Compute the inverse transform at a single point:

In[1]:=1
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-xhwnbt
Out[1]=1

Scope  (43)Survey of the scope of standard use cases

Basic Uses  (3)

Inverse Fourier cosine transform of a function for a symbolic parameter :

In[1]:=1
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-3tfh8t
Out[1]=1

Inverse Fourier cosine transforms involving trigonometric functions:

In[1]:=1
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-i6lofd
Out[1]=1

Plot the transform:

In[2]:=2
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-1b4ghq
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-1mrlz2
Out[3]=3

Plot the transform for :

In[4]:=4
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-guxs6c
Out[4]=4

Evaluate the inverse Fourier cosine transform for a numerical value of the parameter :

In[1]:=1
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-xvwdlm
Out[1]=1

Algebraic Functions  (4)

Inverse Fourier cosine transform of power functions:

In[1]:=1
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-03wsfk
Out[1]=1

For integer , the result is a derivative of DiracDelta:

In[2]:=2
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-vpq0lw
Out[2]=2

Inverse cosine transforms for rational functions:

In[1]:=1
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-owg679
Out[1]=1

Plot the transform:

In[2]:=2
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-0m3rlo
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-boyow9
Out[3]=3

Plot the transform:

In[4]:=4
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-zjzb5g
Out[4]=4

Inverse Fourier cosine transform of a quotient of two nonlinear polynomials:

In[1]:=1
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-rhkhqy
Out[1]=1

Plot the transform:

In[2]:=2
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-i0umyu
Out[2]=2

Inverse Fourier cosine transform of a quotient of quadratic and quartic polynomials:

In[1]:=1
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-jk9mm7
Out[1]=1

Plot the transform:

In[2]:=2
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-z67hdo
Out[2]=2

Exponential and Logarithmic Functions  (4)

Inverse Fourier cosine transforms of exponential functions:

In[1]:=1
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-52om8r
Out[1]=1

Transform for :

In[2]:=2
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-1disjo
Out[2]=2

Plot the transform:

In[3]:=3
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-wu1agn
Out[3]=3

Inverse Fourier cosine transform of a Gaussian is itself:

In[1]:=1
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-yxcp1n
Out[1]=1

Plot the transform:

In[2]:=2
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-o1ry2p
Out[2]=2

Inverse Fourier cosine transforms of products of exponential and trigonometric functions:

In[1]:=1
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-gygzn3
Out[1]=1

Plot the transform for and :

In[2]:=2
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-sxteev
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-jw3ubt
Out[3]=3

Plot the transform:

In[4]:=4
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-3pnitz
Out[4]=4

Inverse cosine transforms of logarithmic functions:

In[1]:=1
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-rgdsei
Out[1]=1

Plot the transform:

In[2]:=2
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-td0odr
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-w54qvj
Out[3]=3

Plot the transform:

In[4]:=4
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-f7hcbj
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-xhk2s9
Out[5]=5

Plot the transform:

In[6]:=6
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-x5had8
Out[6]=6
In[7]:=7
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-8db6mb
Out[7]=7

Plot the transform:

In[8]:=8
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-dhseml
Out[8]=8

Trigonometric Functions  (5)

Expressions involving trigonometric functions:

In[1]:=1
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-i8awco
Out[1]=1

Plot the transform:

In[2]:=2
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-lsavef
Out[2]=2

Composition of elementary functions:

In[1]:=1
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-je3yjb
Out[1]=1

Plot the transform for :

In[2]:=2
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-joqlym
Out[2]=2

Ratio of sine and product of exponential and linear functions:

In[1]:=1
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-e6j8h1
Out[1]=1

Plot the transform:

In[2]:=2
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-oosste
Out[2]=2

Inverse Fourier cosine transforms of arctangent functions:

In[1]:=1
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-5g6245
Out[1]=1

Plot the transform:

In[2]:=2
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-i55usn
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-hz75d7
Out[3]=3

Plot the transform:

In[4]:=4
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-xxlo3y
Out[4]=4

Inverse Fourier cosine transform of Sech is another Sech:

In[1]:=1
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-73qucu
Out[1]=1

Plot the transform:

In[2]:=2
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-09awln
Out[2]=2

Special Functions  (9)

Sinc function:

In[1]:=1
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-0gsyhj
Out[1]=1

Plot the transform:

In[2]:=2
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-vo1vn2
Out[2]=2

Inverse Fourier cosine transforms of expressions involving ExpIntegralEi:

In[1]:=1
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-tjtvu1
Out[1]=1

Plot the transform:

In[2]:=2
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-cwbop2
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-cg821z
Out[3]=3

Plot the transform:

In[4]:=4
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-j66x5w
Out[4]=4

Expression involving Erfc:

In[1]:=1
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-vn2eoq
Out[1]=1

Plot the transform for :

In[2]:=2
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-m91upc
Out[2]=2

Expression involving SinIntegral:

In[1]:=1
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-005pa8
Out[1]=1

Plot the transform:

In[2]:=2
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-7j47yl
Out[2]=2

CosIntegral:

In[1]:=1
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-yznez4
Out[1]=1

Plot the transform:

In[2]:=2
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-ujv0fk
Out[2]=2

Inverse cosine transforms for BesselJ functions:

In[1]:=1
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-gfwe8v
Out[1]=1

Plot the transform for :

In[2]:=2
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-fzv48
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-25dsem
Out[3]=3

Plot the transform for and :

In[4]:=4
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-5ikc3a
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-pu9nv5
Out[5]=5

Plot the transform for and :

In[6]:=6
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-oujib3
Out[6]=6
In[7]:=7
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-0udjyo
Out[7]=7

Plot the transform:

In[8]:=8
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-ucezzb
Out[8]=8

Cosine transforms for BesselY functions:

In[1]:=1
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-jfd8tm
Out[1]=1

Plot the transform for :

In[2]:=2
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-f8qei2
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-u2q3vd
Out[3]=3

Plot the transform for and :

In[4]:=4
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-z955xt
Out[4]=4

Cosine transform for a BesselK function:

In[1]:=1
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-h3rjaf
Out[1]=1

Plot the transform:

In[2]:=2
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-2xs9p4
Out[2]=2

Inverse cosine transform for a hypergeometric function is a BesselK function:

In[1]:=1
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-z8l7t
Out[1]=1

Plot the transform:

In[2]:=2
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-9br70g
Out[2]=2

Piecewise Functions and Distributions  (4)

Inverse Fourier cosine transform of a piecewise function:

In[1]:=1
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-vbyfpc
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-gzj53a
Out[3]=3

Restriction of a sine function to a half-period:

In[1]:=1
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-k1cfgv
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-trakg
Out[2]=2

Triangular function:

In[1]:=1
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-oa5djc
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-osuoi6
Out[2]=2

Transforms in terms of FresnelC:

In[1]:=1
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-3vutk1
Out[1]=1

Plot the transform:

In[2]:=2
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-fye5aq
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-u5hicw
Out[3]=3

Plot the transform:

In[4]:=4
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-l8c2cs
Out[4]=4

Periodic Functions  (2)

Inverse Fourier cosine transform of cosine:

In[1]:=1
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-2zaqf7
Out[1]=1

Inverse Fourier cosine transform of SquareWave:

In[1]:=1
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-sqc6g2
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-58nn6p
Out[2]=2

Generalized Functions  (4)

Inverse Fourier cosine transforms of expressions involving HeavisideTheta:

In[1]:=1
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-2tevp7
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-d1wzw8
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-wyvw1c
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-5fwoi9
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-wqr4mf
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-2y7g21
Out[6]=6

Inverse Fourier cosine transform involving DiracDelta:

In[1]:=1
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-r31kfb
Out[1]=1

Plot the transform:

In[2]:=2
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-nz7s7k
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-wwl2jr
Out[3]=3

Plot the transform:

In[4]:=4
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-qk22x4
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-dq9yjh
Out[5]=5

Plot the transform:

In[6]:=6
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-rowycp
Out[6]=6

Inverse Fourier cosine transform involving HeavisideLambda:

In[1]:=1
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-o6dexh
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-v8ub3l
Out[2]=2

Inverse Fourier cosine transform involving HeavisidePi:

In[1]:=1
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-venbbr
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-zqm430
Out[2]=2

Multivariate Functions  (3)

Inverse Fourier cosine transform of rational function in two variables:

In[1]:=1
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-qs7sar
Out[1]=1

Plot of both:

In[2]:=2
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-yq48d0
Out[2]=2

Inverse Fourier cosine transform of exponential in two variables:

In[1]:=1
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-ibiugp
Out[1]=1

Plot of both:

In[2]:=2
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-oifrh3
Out[2]=2

Inverse Fourier cosine transform of product of exponential and SquareWave:

In[1]:=1
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-xn92xb
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-szbms4
Out[2]=2

Formal Properties  (3)

Inverse Fourier cosine transform of a first-order derivative:

In[1]:=1
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-6y33or
Out[1]=1

Inverse Fourier cosine transform of a second-order derivative:

In[1]:=1
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-skw3qo
Out[1]=1

Inverse Fourier cosine transform threads itself over equations:

In[1]:=1
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-iopmwm
Out[1]=1

Numerical Evaluation  (2)

Calculate the inverse Fourier cosine transform at a single point:

In[1]:=1
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-7k0h22
Out[1]=1

Alternatively, calculate the Fourier cosine transform symbolically:

In[1]:=1
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-tjwigl
Out[1]=1

Then evaluate it for specific value of :

In[2]:=2
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-7r1zof
Out[2]=2

Options  (8)Common values & functionality for each option

AccuracyGoal  (1)

The option AccuracyGoal sets the number of digits of accuracy:

In[1]:=1
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-5ymjb
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-f6u3ua
Out[2]=2

With default settings:

In[3]:=3
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-0fses
Out[3]=3

Assumptions  (1)

Use Assumptions to indicate the region of interest for the parameters:

In[1]:=1
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-nagwk1
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-fk9tc3
Out[2]=2

FourierParameters  (3)

Inverse Fourier cosine transform for the unit box function with different parameters:

Use a nondefault setting for a different definition of transform:

In[1]:=1
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-ejscab
Out[1]=1

To get the original function back, use the same FourierParameters setting:

In[2]:=2
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-x9j3p
Out[2]=2

Set up your particular global choice of parameters to work once per session:

In[1]:=1
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-p5uq9c
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-qntpci

Restore defaults:

In[3]:=3
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-8p4bec
Out[3]=3

GenerateConditions  (1)

Use GenerateConditionsTrue to get parameter conditions for when a result is valid:

In[1]:=1
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-dosv41
Out[1]=1

PrecisionGoal  (1)

The option PrecisionGoal sets the relative tolerance in the integration:

In[1]:=1
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-n3x5d0
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-9119x
Out[2]=2

With default settings:

In[3]:=3
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-b3dg8e
Out[3]=3

WorkingPrecision  (1)

If a WorkingPrecision is specified, the computation is done at that working precision:

In[1]:=1
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-c6xb9r
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-duvdjv
Out[2]=2

With default settings:

In[3]:=3
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-f3d1p1
Out[3]=3

Applications  (4)Sample problems that can be solved with this function

Ordinary Differential Equations  (1)

Consider the following ODE with initial condition :

In[1]:=1
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-7nyko5

Apply the Fourier cosine transform to the ODE:

In[2]:=2
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-vneovh
Out[2]=2

Solve for the Fourier cosine transform of :

In[3]:=3
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-o8i7eo
Out[3]=3

Find the inverse Fourier cosine transform with and :

In[4]:=4
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-o2vnsc
Out[4]=4

Compare with DSolveValue:

In[5]:=5
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-p6a0xe
Out[5]=5

Partial Differential Equations  (1)

Solve the heat equation for , : with initial condition for and Neumann boundary condition for :

In[1]:=1
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-px64a0

Apply the Fourier cosine transform to the ODE on :

In[2]:=2
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-2kme07
Out[2]=2

With and , solve this ODE:

In[3]:=3
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-ehtvs7
Out[3]=3

Compute the inverse cosine transform of the exponential functions:

In[4]:=4
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-ccx48j

Convolution property gives the inverse cosine transform of the first summand to get the solution:

In[6]:=6
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-h6hd7
Out[6]=6

Consider the special case with , and :

In[7]:=7
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-jgjd13
Out[7]=7

Compare with DSolveValue:

In[8]:=8
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-dkim4w
Out[8]=8

Plot the initial conditions and solutions for different values of .

In[9]:=9
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-t3wb9i
Out[9]=9

Evaluation of Integrals  (2)

Calculate the following definite integral:

In[1]:=1
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-mvl6f4
Out[1]=1

Inverse Fourier cosine transform preserves integration of products over :

In[2]:=2
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-0i61ex
Out[2]=2

Solve the definite integral:

In[3]:=3
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-gojuf4
Out[3]=3

Compare with Integrate:

In[4]:=4
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-8yvy61
Out[4]=4

Calculate the following definite integral for :

In[1]:=1
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-xpx1az
Out[1]=1

Compute inverse fourier cosine transform of the square root of the integrand:

In[2]:=2
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-g76rkm
Out[2]=2

Apply Parseval's identity:

In[3]:=3
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-l3jwfg
Out[3]=3

Or equivalently:

In[4]:=4
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-5w5yeo
Out[4]=4

Solve for the definite integral:

In[5]:=5
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-y2ew8h
Out[5]=5

Compare with Integrate:

In[6]:=6
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-e84l8t
Out[6]=6

Properties & Relations  (4)Properties of the function, and connections to other functions

By default, the inverse Fourier cosine transform of is:

In[1]:=1
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-hskcwe

For , the definite integral becomes:

In[2]:=2
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-12m121
Out[2]=2

Compare with InverseFourierCosTransform:

In[3]:=3
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-zlwqzc
Out[3]=3

Use Asymptotic to compute an asymptotic approximation:

In[1]:=1
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-lwbzos
Out[1]=1

FourierCosTransform and InverseFourierCosTransform are mutual inverses:

In[1]:=1
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-bp78fy
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-bhj9ka
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-it7bfn
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-byd6ra
Out[4]=4

For even functions results are identical to InverseFourierTransform:

In[1]:=1
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-juanwx
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-ib6ihh
Out[2]=2

The results agree for :

In[3]:=3
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-b2ebmr
Out[3]=3

Possible Issues  (1)Common pitfalls and unexpected behavior

The result from a Fourier cosine transform may not have the same form as the original:

In[1]:=1
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-jod215
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-19v81k
Out[2]=2

Inverse Fourier cosine transforms may require generalized functions such as DiracDelta:

In[3]:=3
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-mjrqa7
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-f2qo5e
Out[4]=4

Neat Examples  (2)Surprising or curious use cases

Inverse Fourier cosine transform as a Meijer function:

In[1]:=1
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-pqr5v5
Out[1]=1

Create a table of basic inverse Fourier cosine transforms:

In[1]:=1
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-m5ymd1
In[2]:=2
✖
https://wolfram.com/xid/0ixfgt46t1bkc5e-7mblzr
Wolfram Research (1999), InverseFourierCosTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/InverseFourierCosTransform.html (updated 2025).
✖
Wolfram Research (1999), InverseFourierCosTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/InverseFourierCosTransform.html (updated 2025).

Text

Wolfram Research (1999), InverseFourierCosTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/InverseFourierCosTransform.html (updated 2025).

✖
Wolfram Research (1999), InverseFourierCosTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/InverseFourierCosTransform.html (updated 2025).

CMS

Wolfram Language. 1999. "InverseFourierCosTransform." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2025. https://reference.wolfram.com/language/ref/InverseFourierCosTransform.html.

✖
Wolfram Language. 1999. "InverseFourierCosTransform." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2025. https://reference.wolfram.com/language/ref/InverseFourierCosTransform.html.

APA

Wolfram Language. (1999). InverseFourierCosTransform. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/InverseFourierCosTransform.html

✖
Wolfram Language. (1999). InverseFourierCosTransform. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/InverseFourierCosTransform.html

BibTeX

@misc{reference.wolfram_2025_inversefouriercostransform, author="Wolfram Research", title="{InverseFourierCosTransform}", year="2025", howpublished="\url{https://reference.wolfram.com/language/ref/InverseFourierCosTransform.html}", note=[Accessed: 11-July-2025 ]}

✖
@misc{reference.wolfram_2025_inversefouriercostransform, author="Wolfram Research", title="{InverseFourierCosTransform}", year="2025", howpublished="\url{https://reference.wolfram.com/language/ref/InverseFourierCosTransform.html}", note=[Accessed: 11-July-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_inversefouriercostransform, organization={Wolfram Research}, title={InverseFourierCosTransform}, year={2025}, url={https://reference.wolfram.com/language/ref/InverseFourierCosTransform.html}, note=[Accessed: 11-July-2025 ]}

✖
@online{reference.wolfram_2025_inversefouriercostransform, organization={Wolfram Research}, title={InverseFourierCosTransform}, year={2025}, url={https://reference.wolfram.com/language/ref/InverseFourierCosTransform.html}, note=[Accessed: 11-July-2025 ]}